New Problems in Mean Field Control Theory

平均场比赛是在2006年由Lasry-Lions和Huang-Caines-Malhamé独立推出的。他们的想法是将平均场方法从物理和力学应用到社会科学。平均场方法允许描述流体和气体的宏观行为，而不考虑单个粒子的运动。我们可以考虑一个具有代表性的粒子，它的动力学受到一个平均场项的影响，该平均场项表示来自所有其他粒子的平均效应。在社会科学中，粒子被试剂取代。例如，在对高速公路上的交通进行建模时，代理可以是一辆汽车。与物理学的一个重要区别是，代理人是决策者，动力学与决策相对应。这项提议为这些平均场模型开发了控制方法，称为平均场控制理论。平均场控制理论的一个基本问题是求解Hamilton-Jacobi-Bellman和Fokker Planck(HJB-FP)方程组。由狮子座提出的主方程，允许人们通过驱动解耦论点将HJB-FP方程系统封装在单个方程中。一个严重的困难是，人们需要使用概率度量的函数。狮子座提出了这样一种想法，即人们可以用平方可积随机变量的希尔伯特空间来写出主方程。这导致了在这一特殊的希尔伯特空间中出现新的控制问题，这是本研究的第一个目标。第二个目标是研究平均场控制理论中的具体问题，这是出于经济学的动机。

Mean field games were introduced in 2006, independently by Lasry-Lions and Huang-Caines-Malhamé. Their idea was to adapt the mean field approach from physics and mechanics to social sciences. The mean field approach allows the description of the macroscopic behavior of fluids and gases without considering the motion of the individual particles. One can consider a representative particle whose dynamics is influenced by a mean field term that represents an averaging effect coming from all other particles. In the social sciences, particles are replaced with agents. For example, an agent can be a car when modeling traffic on highways. An important difference from physics is that the agents are decision makers and the dynamics correspond to decisions. This proposal develops control methods for these mean field models, called Mean Field Control Theory. Results will be applied to dynamic networks and provide insight for applications such as economic theory.A basic problem in mean field control theory is to solve a system of Hamilton-Jacobi-Bellman and Fokker Planck (HJB-FP) equations. The Master Equation, introduced by Lions, allows one to encapsulate the system of HJB-FP equations in a single equation by driving a decoupling argument. A serious difficulty is the fact that one needs to work with functions of probability measures. Lions introduced the idea that one could write the Master Equation using for the space of arguments the Hilbert space of square integrable random variables. This leads to new control problems in this particular Hilbert space, which is the first objective of the research. The second objective is to study specific problems in mean field control theory, motivated by economics.